The composed functors \(\mathcal {T}_k:=\mbox{Tr}\,\Omega ^{k-1}\)  for \(k>0\)  introduced by Auslander and Bridger in [1]}. If \(\mbox{Ext}\,^i_R(M,R)=0\)  for some \(i>0\) , then it is easy to see that \(\underline{\mathcal {T}_iM}\cong \underline{\Omega \mathcal {T}_{i+1}M}\) .
